Fourier Transform (Chapter 4) - PowerPoint PPT Presentation

About This Presentation
Title:

Fourier Transform (Chapter 4)

Description:

Mathematical Background: Complex Numbers (cont'd) Euler's formula. Properties. j. Mathematical Background: Sine and Cosine Functions ... – PowerPoint PPT presentation

Number of Views:97
Avg rating:3.0/5.0
Slides: 56
Provided by: george76
Learn more at: https://www.cse.unr.edu
Category:

less

Transcript and Presenter's Notes

Title: Fourier Transform (Chapter 4)


1
Fourier Transform (Chapter 4)
  • CS474/674 Prof. Bebis

2
Mathematical BackgroundComplex Numbers
  • A complex number x is of the form
  • a real part, b imaginary
    part
  • Addition
  • Multiplication

3
Mathematical BackgroundComplex Numbers (contd)
  • Magnitude-Phase (i.e., vector) representation

  • Magnitude
  • Phase

f
Magnitude-Phase notation
4
Mathematical BackgroundComplex Numbers (contd)
  • Multiplication using magnitude-phase
    representation
  • Complex conjugate
  • Properties

5
Mathematical BackgroundComplex Numbers (contd)
  • Eulers formula
  • Properties

6
Mathematical BackgroundSine and Cosine Functions
  • Periodic functions
  • General form of sine and cosine functions

y(t)Asin(atb) y(t)Acos(atb)
7
Mathematical BackgroundSine and Cosine Functions
Special case A1, b0, a1
period2p
p
3p/2
p/2
p
3p/2
p/2
8
Mathematical BackgroundSine and Cosine
Functions (contd)
  • Changing the phase shift b

Note cosine is a shifted sine function
9
Mathematical BackgroundSine and Cosine
Functions (contd)
  • Changing the amplitude A

10
Mathematical BackgroundSine and Cosine
Functions (contd)
  • Changing the period T2p/a
  • Asssume A1, b0 ycos(at)

a 4
period 2p/4p/2
shorter period higher frequency (i.e.,
oscillates faster)
frequency is defined as f1/T
Alternative notation cos(at) or cos(2pt/T) or
cos(t/T) or cos(2pft) or cos(ft)
11
Basis Functions
  • Given a vector space of functions, S, then if any
    f(t) ? S can be expressed as
  • the set of functions fk(t) are called the
    expansion set of S.
  • If the expansion is unique, the set fk(t) is a
    basis.

12
Image Transforms
  • Many times, image processing tasks are best
    performed in a domain other than the spatial
    domain.
  • Key steps
  • (1) Transform the image
  • (2) Carry the task(s) in the transformed domain.
  • (3) Apply inverse transform to return to the
    spatial domain.

13
Transformation Kernels
forward transformation kernel
  • Forward Transformation
  • Inverse Transformation

inverse transformation kernel
14
Kernel Properties
  • A kernel is said to be separable if
  • A kernel is said to be symmetric if

15
Fourier Series Theorem
  • Any periodic function f(t) can be expressed as a
    weighted sum (infinite) of sine and cosine
    functions of varying frequency

is called the fundamental frequency
 
 
16
Fourier Series (contd)
a1
a2
a3
17
Continuous Fourier Transform (FT)
  • Transforms a signal (i.e., function) from the
    spatial (x) domain to the frequency (u) domain.

where
18
Definitions
  • F(u) is a complex function
  • Magnitude of FT (spectrum)
  • Phase of FT
  • Magnitude-Phase representation
  • Power of f(x) P(u)F(u)2

19
Why is FT Useful?
  • Easier to remove undesirable frequencies in the
    frequency domain.
  • Faster to perform certain operations in the
    frequency domain than in the spatial domain.

20
Example Removing undesirable frequencies
frequencies
noisy signal
remove high frequencies
reconstructed signal
To remove certain frequencies, set
their corresponding F(u) coefficients to zero!
21
How do frequencies show up in an image?
  • Low frequencies correspond to slowly varying
    pixel intensities (e.g., continuous surface).
  • High frequencies correspond to quickly varying
    pixel intensities (e.g., edges)

Original Image
Low-passed
22
Example of noise reduction using FT
Input image
Spectrum (frequency domain)
Output image
Band-reject filter
23
Frequency Filtering Main Steps
  • 1. Take the FT of f(x)
  • 2. Remove undesired frequencies
  • 3. Convert back to a signal

Well talk more about these steps later .....
24
Example rectangular pulse
rect(x) function
sinc(x)sin(x)/x
25
Example impulse or delta function
  • Definition of delta function
  • Properties

 
26
Example impulse or delta function (contd)
  • FT of delta function

27
Example spatial/frequency shifts

Special Cases
28
Example sine and cosine functions
  • FT of the cosine function

cos(2pu0x)
F(u)
1/2
29
Example sine and cosine functions (contd)
  • FT of the sine function

 
jF(u)
sin(2pu0x)
30
Extending FT in 2D
  • Forward FT
  • Inverse FT

31
Example 2D rectangle function
  • FT of 2D rectangle function

2D sinc()
top view
32
Discrete Fourier Transform (DFT)
33
Discrete Fourier Transform (DFT) (contd)
  • Forward DFT
  • Inverse DFT

34
Example
35
Extending DFT to 2D
  • Assume that f(x,y) is M x N.
  • Forward DFT
  • Inverse DFT

36
Extending DFT to 2D (contd)
  • Special case f(x,y) is N x N.
  • Forward DFT
  • Inverse DFT

u,v 0,1,2, , N-1
x,y 0,1,2, , N-1
37
Extending DFT to 2D (contd)
2D cos/sin functions
Interpretation
38
Visualizing DFT
  • Typically, we visualize F(u,v)
  • The dynamic range of F(u,v) is typically very
    large
  • Apply streching
    (c is const)

D(u,v)
F(u,v)
original image
before stretching
after stretching
39
DFT Properties (1) Separability
  • The 2D DFT can be computed using 1D transforms
    only
  • Forward DFT

kernel is separable
40
DFT Properties (1) Separability (contd)
  • Rewrite F(u,v) as follows
  • Lets set
  • Then

41
DFT Properties (1) Separability (contd)
  • How can we compute F(x,v)?
  • How can we compute F(u,v)?

N x DFT of rows of f(x,y)
DFT of cols of F(x,v)
42
DFT Properties (1) Separability (contd)
43
DFT Properties (2) Periodicity
  • The DFT and its inverse are periodic with period
    N

44
DFT Properties (3) Symmetry
45
DFT Properties (4) Translation
  • Translation in spatial domain
  • Translation in frequency domain

46
DFT Properties (4) Translation (contd)
  • To show a full period, we need to translate the
    origin of the transform at uN/2 (or at (N/2,N/2)
    in 2D)

47
DFT Properties (4) Translation (contd)
  • To move F(u,v) at (N/2, N/2), take

48
DFT Properties (4) Translation (contd)
sinc
sinc
no translation
after translation
49
DFT Properties (5) Rotation
  • Rotating f(x,y) by ? rotates F(u,v) by ?

50
DFT Properties (6) Addition/Multiplication
but
51
DFT Properties (7) Scale
52
DFT Properties (8) Average value
Average
F(u,v) at u0, v0
So
53
Magnitude and Phase of DFT
  • What is more important?
  • Hint use the inverse DFT to reconstruct the
    input image using only magnitude or phase
    information

magnitude
phase
54
Magnitude and Phase of DFT (contd)
Reconstructed image using magnitude only (i.e.,
magnitude determines the strength of each
component)
Reconstructed image using phase only (i.e.,
phase determines the phase of each component)
55
Magnitude and Phase of DFT (contd)
only phase
only magnitude
phase (woman) magnitude (rectangle)
phase (rectangle) magnitude (woman)
Write a Comment
User Comments (0)
About PowerShow.com